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Inequality (Posted on 2015-02-11)

Show that (sin x)^(sin x) < (cos x)^(cos x) when 0 < x < pi/4

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

Comment 5 of 5 |

Using the power series   log_e(1 - t) = – t - t²/2 – t³/3 …. and

the identity cos²x = 1 – sin²x :

            log_e(cos²x) = -sin²x – sin⁴x/2 – sin⁶x/3 -…

Thus     2*log_e(c) = -s² – s⁴/2 – s⁶/3 …     (s & c for sinx and cosx)

            2c*log_e(c) = -sc(s + s³/2 + s⁵/2 +…)                     (1)

Now, starting with sin²x = 1 – cos²x, the same procedure gives:

            2s*log_e(s) = -cs(c + c³/2 + c⁵/3 + …)                    (2)

(2) – (1) now gives:

2[s*log_e(s) – c*log_e(c)] = sc[(s – c) + (s³ – c³)/2 + (s⁵ – c⁵)/3 …

For 0 < x < pi/4, 0 < s < c < 1, so the RHS is clearly negative,

and therefore    s*log_e(s) < c*log_e(c),

which gives:   log_e(s^s) < log_e(c^c) and   s^s < c^c as required.

I’m hoping someone can find an easier way than this..

Posted by Harry on 2015-02-15 14:54:25

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